Advantages of using relative-phase Toffoli gates with an application to multiple control Toffoli optimization
Open accessAbstract
Various implementations of the Toffoli gate up to a relative phase have been known for years. The advantage over the regular Toffoli gate is its smaller circuit size. However, its use has been often limited to a demonstration of quantum control in designs such as those where the Toffoli gate is being applied last or otherwise for some specific reasons the relative phase does not matter. It was commonly believed that the relative-phase deviations would prevent the relative-phase Toffoli gates from being very helpful in practical large-scale designs. In this paper, we report three circuit identities that provide the means for replacing certain configurations of the multiple control Toffoli gates with their simpler relative-phase implementations, up to a selectable unitary on certain qubits, and without changing the overall functionality. We illustrate the advantage via applying those identities to the optimization of the known circuits implementing multiple control Toffoli gates, and report the reductions in the controlled-not count, $T$count, as well as the number of ancillae used. We suggest that a further study of the relative-phase Toffoli implementations and their use may yield other optimizations.
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The AND operation can be implemented with only 3 CNOTs using a relative phase Toffoli implementation given by Qiskit's Margolus gate RCCX and a consecutive S†.
“The AND can be implemented with only 3 CNOTs using a relative phase Toffoli implementation [36], [54] given by Qiskit's Margolus gate RCCX and a consecutive S † .”
From High-Round QAOA for MAX $k$-SAT on Trapped Ion NISQ Devices (2023) · cites this paper as [36]
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“11 Box plots for relative difference between input image [10, 85, 170, 255] and output image from five different backends using the MARY-implementation.”
BackgroundMethodInfluentialVariational Quantum Shot-Based Simulations for Waveguide Modes
Colella, E., Beloin, S., Bastianelli, L., Primiani, V. M., Moglie, F., & Gradoni, G. (2023) · IEEE Transactions on Microwave Theory and Techniques · doi:10.1109/tmtt.2023.3339243 · Cited by 15 · see citing works
Compiling Permutations for Superconducting QPUs
Soeken, M., Mozafari, F., Schmitt, B. D. O., & Micheli, G. D. (2019) · Design, Automation, and Test in Europe · doi:10.23919/date.2019.8715275 · Cited by 15 · see citing works
“Finally, each remaining multiple-controlled Toffoli gate is decomposed into a Clifford+T circuit as described in [19].”
MethodOonishi, K., Tanaka, T., Uno, S., Satoh, T., Meter, R. V., & Kunihiro, N. (2021) · IEEE Transactions on Quantum Engineering · doi:10.1109/tqe.2021.3136195 · Cited by 17 · see citing works
High-Round QAOA for MAX $k$-SAT on Trapped Ion NISQ Devices
Pelofske, E., Bärtschi, A., Golden, J., & Eidenbenz, S. (2023) · 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) · doi:10.1109/qce57702.2023.00064 · Cited by 15 · see citing works
Riener, H., Ehlers, R., Schmitt, B., & Micheli, G. D. (2018) · Advanced Boolean Techniques · doi:10.1007/978-3-030-20323-8_8 · Cited by 23 · see citing works
Simulating lattice quantum electrodynamics on a quantum computer
Kan, A., & Nam, Y. (2022) · Quantum Science and Technology · doi:10.1088/2058-9565/aca0b8 · Cited by 13 · see citing works
Implementing the Grover algorithm in homomorphic encryption schemes
Fernández, P., & Martin-Delgado, M. A. (2024) · Physical Review Research · doi:10.1103/physrevresearch.6.043109 · Cited by 13 · see citing works
Reversible Pebble Games for Reducing Qubits in Hierarchical Quantum Circuit Synthesis
Bhattacharjee, D., Soeken, M., Dutta, S., Chattopadhyay, A., & Micheli, G. D. (2019) · IEEE International Symposium on Multiple-Valued Logic · doi:10.1109/ismvl.2019.00026 · Cited by 10 · see citing works
Liu, H.-T., Chen, B., Zhang, J., Xiao, Y.-X., Li, T., Huang, K., Wang, Z., Li, H., Zhao, K., Xu, Y., et al. (2025) · Physical Review Letters · doi:10.1103/lvb9-pfr3 · Cited by 12 · see citing works
Quantum circuits for SU(3) lattice gauge theory
Balaji, P., Conefrey-Shinozaki, C., Draper, P., Elhaderi, J. K., Gupta, D., Hidalgo, L., Lytle, A., & Rinaldi, E. (2025) · Physical review. D/Physical review. D. · doi:10.1103/k8f6-yft8 · Cited by 12 · see citing works
Uncomputation in the Qrisp High-Level Quantum Programming Framework
Seidel, R., Tcholtchev, N., Bock, S., & Hauswirth, M. (2023) · Lecture notes in computer science · doi:10.1007/978-3-031-38100-3_11 · Cited by 12 · see citing works
Finding Solutions to the Integer Case Constraint Satisfiability Problem Using Grover’s Algorithm
Vinod, G. M., & Shaji, A. (2021) · IEEE Transactions on Quantum Engineering · doi:10.1109/tqe.2021.3120449 · Cited by 12 · see citing works
Applying Grover's algorithm to AES: quantum resource estimatesPreprint
Grassl, M., Langenberg, B., Roetteler, M., & Steinwandt, R. (2015) · arXiv (Cornell University) · doi:10.48550/arxiv.1512.04965 · Cited by 12 · see citing works
Handling Non-unitaries in Quantum Circuit Equivalence Checking
Burgholzer, L., & Wille, R. (2022) · Springer eBooks · doi:10.1007/978-3-032-06770-8_18 · Cited by 14 · see citing works
Linear-Scaling Quantum Circuits for Computational Chemistry
Magoulas, I., & Evangelista, F. A. (2023) · Journal of Chemical Theory and Computation · doi:10.1021/acs.jctc.3c00376 · Cited by 11 · see citing works
Improved NCV Gate Realization of Arbitrary Size Toffoli Gates
Kole, A., & Datta, K. (2017) · International Conference on VLSI Design · doi:10.1109/vlsid.2017.11 · Cited by 11 · see citing works
“Realizing the operation of a Toffoli gate using Clifford+T gate library has been reported in [10].”
MethodSignal processing techniques for efficient compilation of controlled rotations in trapped ions
Groenland, K., Witteveen, F., Schoutens, K., & Gerritsma, R. (2020) · New Journal of Physics · doi:10.1088/1367-2630/ab8830 · Cited by 12 · see citing works
The T-Complexity Costs of Error Correction for Control Flow in Quantum Computation
Yuan, C., & Carbin, M. (2024) · Proceedings of the ACM on Programming Languages · doi:10.1145/3656397 · Cited by 10 · see citing works
A Quantum Algorithm for System Specifications Verification
Zidan, M., Eisa, A. M., Qasymeh, M., & Shoman, M. A. I. (2024) · IEEE Internet of Things Journal · doi:10.1109/jiot.2024.3383034 · Cited by 10 · see citing works
Automatic Uniform Quantum State Preparation Using Decision Diagrams
Mozafari, F., Soeken, M., Riener, H., & Micheli, G. D. (2020) · IEEE International Symposium on Multiple-Valued Logic (ISMVL) · doi:10.1109/ismvl49045.2020.00-10 · Cited by 10 · see citing works
Meuli, G., Soeken, M., Roetteler, M., Wiebe, N., & Micheli, G. D. (2018) · Asia and South Pacific Design Automation Conference · doi:10.1109/aspdac.2018.8297398 · Cited by 13 · see citing works
Linear and Non-linear Relational Analyses for Quantum Program Optimization
Amy, M., & Lunderville, J. (2025) · Proceedings of the ACM on Programming Languages · doi:10.1145/3704873 · Cited by 9 · see citing works
- Authors
- Maslov, D.
- Affiliations
- U.S. National Science Foundation (US)
- Corresponding author
- Maslov, Dmitri
- ORCID
- Maslov, Dmitri
- Venue
- Physical Review A
- Publisher
- American Physical Society (APS)
- Published
- 2016-02-10
- Volume
- 93
- Issue
- 2
- ISSN
- 0556-2791 (ISSN-L)
- 2469-9926
- 2469-9934
- OpenAlex
- W1749786511
- Cited by
- 193
- References
- 11
- Citations by year
- 2026: 17 · 2025: 19 · 2024: 28 · 2023: 23 · 2022: 30 · 2021: 23 · 2020: 17 · 2019: 12 · 2018: 8 · 2017: 10 · 2016: 5 · 2015: 1
- FWCI
- 9.61
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Licence
- Locations
- Physical Review A (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- National Science Foundation
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Quantum-Dot Cellular Automata
- Keywords
- quantum circuit synthesis
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